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Square free values of the order function. (English) Zbl 1066.11044

Summary: Given \(a\in\mathbb{Z}\setminus\{\pm1,0\}\), we consider the problem of enumerating the integers \(m\) coprime to \(a\) such that the order of \(a\) modulo \(m\) is square free. This question is raised in analogy to a result recently proved jointly with F. Saidak and I. Shparlinski where square free values of the Carmichael function are studied. The technique is the one of Hooley that uses the Chebotarev Density Theorem to enumerate primes for which the index \(i_p(a)\) of \(a\) modulo \(p\) is divisible by a given integer.

MSC:

11N37 Asymptotic results on arithmetic functions
11N56 Rate of growth of arithmetic functions
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